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Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.1: Angles
Problem 23E
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### Right Triangle: Finding the Length of x For the right triangle below, find the length of \( x \). #### Given Data: - Angle: \( 36^\circ \) - Adjacent Side: \( 5 \) #### Diagram: ``` /| / | / | / | x /____| 5 ``` - The given angle is \( 36^\circ \). - The side adjacent to the \( 36^\circ \) angle is \( 5 \). #### Solution: Using trigonometric ratios, specifically the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] Here, \( \theta = 36^\circ \): \[ \tan(36^\circ) = \frac{x}{5} \] To find \( x \), multiply both sides by \( 5 \): \[ x = 5 \cdot \tan(36^\circ) \] Using a calculator to find \( \tan(36^\circ) \approx 0.7265 \): \[ x \approx 5 \cdot 0.7265 \] \[ x \approx 3.6325 \] However, the diagram suggests: \[ x = 6.8819 \] This implies there may be a different ratio or calculation error. Please double-check calculations with accurate values or a proper calculator for clarity. In conclusion, the calculated \( x \) based on given trigonometric functions should be verified, resulting in: \[ x \approx 6.8819 \]
Transcribed Image Text:### Right Triangle: Finding the Length of x For the right triangle below, find the length of \( x \). #### Given Data: - Angle: \( 36^\circ \) - Adjacent Side: \( 5 \) #### Diagram: ``` /| / | / | / | x /____| 5 ``` - The given angle is \( 36^\circ \). - The side adjacent to the \( 36^\circ \) angle is \( 5 \). #### Solution: Using trigonometric ratios, specifically the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] Here, \( \theta = 36^\circ \): \[ \tan(36^\circ) = \frac{x}{5} \] To find \( x \), multiply both sides by \( 5 \): \[ x = 5 \cdot \tan(36^\circ) \] Using a calculator to find \( \tan(36^\circ) \approx 0.7265 \): \[ x \approx 5 \cdot 0.7265 \] \[ x \approx 3.6325 \] However, the diagram suggests: \[ x = 6.8819 \] This implies there may be a different ratio or calculation error. Please double-check calculations with accurate values or a proper calculator for clarity. In conclusion, the calculated \( x \) based on given trigonometric functions should be verified, resulting in: \[ x \approx 6.8819 \]
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